Classical and quantum motion in an inverse square potential

Abstract

Classical motion in an inverse square potential is shown to be equivalent to free motion on a hyperbola. The existence of a classical splitting between the q > 0 and q < 0 regions of motion is demonstrated. We show that this last property may be regarded as the classical counterpart of the superselection rule occurring in the corresponding quantum problem. We solve the quantum problem in momentum space finding that there is no way of quantizing its energy but that the eigenfunctions suffice to describe the single renormalized bound state of the system. The dynamical symmetry of the classical problem is found to be O(1, 1). Both this symmetry and the symmetry of inversion through the origin are found to be broken. (C) 2008 Elsevier B.V. All rights reserved.